The No Long Odd Cycle Theorem for Completely Positive Matrices

We present a self-contained proof of the following fact. Let an undirected graph G be given. Every symmetric matrix A, with graph G, that is both entry-wise nonnegative and positive semidefinite can be written as A = BBT with B entry-wise nonnegative if and only if G has no odd cycle of length 5 or more. In the process, we determine the worst case for the minimum number of columns in B in the representation of such an A. An n-by-n matrix A = (aij) is called completely positive if A may be written in which B is n-by-m and entry-wise nonnegative. We write A E CP, or C Pn if it is necessary to indicate the dimension. Though they arise in a variety of ways (H), there is yet no definitive test for a matrix to be com­ pletely positive. Recent work has also related CP matrices to exchangeable probability distributions on finite sample spaces (D). A straightforward ob­ servation is that the definition of complete positivity could equivalently be stated as